As the outcome of a collaborative work that integrates research-based practices from mathematics education and special education, the researchers in this project have developed an intelligent tutor, PGBM-COMPS. The PGBM-COMPS intelligent tutor nurtures multiplicative reasoning and problem- solving ability of students with learning disabilities or difficulties in mathematics (LDM). The PGBM-COMPS intelligent tutor draws on three research-based frameworks: a constructivist view of learning from mathematics education (Tzur et al., 2013; Steffe & D’Ambrosio, 1995), data (or statistical) learning from computer sciences (Cetintas, Si, Xin, & Hord, 2009), and Conceptual Model-based Problem Solving (COMPS, Xin, 2012) from special education. The PGBM-COMPS tutor is made of two parts: (1) “Please Go Bring Me …” (PGBM) turn-taking games designed to nurture a learner’s construction of fundamental mathematical ideas in multiplicative reasoning; and (2) COMPS that emphasizes understanding and representation of mathematical relations, decontextualized from real-world problems, in algebraic equations. The PGBM-COMPS tutor generalizes the understanding of multiplicative reasoning to the level of mathematical models and therefore, it promotes generalized problem solving skills. A series of studies have been conducted to test the effectiveness of the PGBM-COMPS program. Findings from empirical studies have shown promising effect of the PGBM-COMPS program for students with LDM in particular.

As the outcome of a collaborative work that integrates research-based practices from mathematics education and special education, the researchers in this project have developed an intelligent tutor, PGBM-COMPS. The PGBM-COMPS intelligent tutor nurtures multiplicative reasoning and problem- solving ability of students with learning disabilities or difficulties in mathematics (LDM). The PGBM-COMPS intelligent tutor draws on three research-based frameworks: a constructivist view of learning from mathematics education (Tzur et al., 2013; Steffe & D’Ambrosio, 1995), data (or statistical) learning from computer sciences (Cetintas, Si, Xin, & Hord, 2009), and Conceptual Model-based Problem Solving (COMPS, Xin, 2012) from special education. The PGBM-COMPS tutor is made of two parts: (1) “Please Go Bring Me …” (PGBM) turn-taking games designed to nurture a learner’s construction of fundamental mathematical ideas in multiplicative reasoning; and (2) COMPS that emphasizes understanding and representation of mathematical relations, decontextualized from real-world problems, in algebraic equations. The PGBM-COMPS tutor generalizes the understanding of multiplicative reasoning to the level of mathematical models and therefore, it promotes generalized problem solving skills. A series of studies have been conducted to test the effectiveness of the PGBM-COMPS program. Findings from empirical studies have shown promising effect of the PGBM-COMPS program for students with LDM in particular.

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Your program will be important for students with learning difficulties and for students who get “stuck” on particular mathematical concepts. What were the demographics of the students in your studies? How did you select which concepts to include in the intelligent tutor? What difficulties or challenges did you encounter technically or conceptually in the development of the tutor? How do you plan to broaden participation and utilization of your tool?

Thanks, Vivian, for all your insightful questions.
This program focuses on fundamental mathematical ideas that is critical to multiplicative reasoning (MR), such as Number as a Composite Unit (CU) and Multiplicative Double Counting (mDC). As such we included tasks such as same unit coordination, unit differentiation and selection, and mixed unit coordination in the PGMB part of the program. After students have established fundamental math ideas that are pertinent to MR, we present students with a range of multiplicative word problems (in the COMPS part of the program) including equal group problems as well as multiplicative compare problems with the unknown in all possible positions to facilitate student’s construction of multiplicative problem schemata.
The most challenging part is for the tutor to provide feedback that will address each individual’s unique misconceptions.
How do you plan to broaden participation and utilization of your tool?
In fact, this program is useful for both under-performing and average performing students at large as it is designed to teach connections among mathematical ideas (i.e., connections between a range of multiplicative problem types) through mathematical model based problem solving, which in turn facilitates generalized problem solving skills.

Meixia Ding

Yanping, I am fascinated by your COMPS! In your model, is there a definite sequence in representation uses? (say, from a real-world context to a mathematics model, and then to a generalized problem solving model). It sounds like the idea of generalized worked example? Also, does Comps stress representation uses such as schematic diagrams?

The intelligent tutor has a set sequence of tasks for students to develop fundamental mathematical ideas and to promote students’ learning from the stage of concrete operation to more advanced abstract level of operation for generalized problem solving skills.
The tutor includes five modules—
See the video link below (a longer version of our project video) for more information— https://www.youtube.com/watch?v=VCc0wjPLXjE

In particular, at time tag 1:56, the video presents the five modules in this program. I hope that answers your question about the task sequences in this intelligent tutor program.
There are specific promotion criteria—which determine which or whether the student needs to work on certain tasks.

COMPS does NOT use schema-based diagrams, instead, it uses cohesive mathematics models that make the connections among numerous schematic diagrams. This is one of the unique features and the advantages of the COMPS program, which benefit students with LDs, in particular, who have limited working memory.

Debra Bernstein

Facilitator

May 11, 2015 | 09:45 p.m.

This sounds like a very interesting tool. Can you say more about the context of use? Is PGBM-COMPS generally used as a stand-alone tool?

Thanks, Debra, for your question.
It will be used primarily as a supplemental curriculum to address students’ knowledge gap and conceptual understanding in multiplicative word problem solving. Nevertheless, it definitely can be used as the instructional program for teaching multiplicative word problem-solving part of the regular elementary curriculum:)

I really like how you are leveraging technology to support student learning! Could you talk more about the range of tasks that are present? What type of feedback does the tutor provide during these tasks, and when is the feedback given?

In particular, at time tag 1:56, the video presents the five modules (including information about the structure of the tasks) in this program. Also please see the two tables attached below for the range of the tasks.

The feedback is designed based on a schefolding scheme: if the student is unable to solve the problem at the abstract level, then the program will bring in the concrete manipulatives (e.g. cubes or towers) to help student understand the concept. The feedback will move from more generic to more specific ones. In short, informative feedback will be given to all student’s actions—whether it is correct or incorrect.
The most challenging part is for the tutor to provide feedback that will address each individual learner’s unique misconceptions.

Table 1
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Sample Tasks across the first Four Modules of the PGBM-COMPS Intelligent System (Purdue Research Foundation, 2011)

Problem Type Sample Problems Situations
Multiplicative Double Counting Pretend that you have made many towers, each made of 7 cubes.
How many cubes are in every tower? Your Answer: _____________
How many cubes are in the first 4 towers? Your Answer: _____________
So we can count those by seven, “7, 14, 21, 28 …”
Do you think you will say the number 70 if you continue counting cubes in the towers?
Your Answer: _____________
Why? _________________________________________________
Do you think you will say the number 84 if you continue counting cubes in the towers?
Your Answer: _____________
Why? _____________________________________________________

Same Unit Coordination Rachael has built 13 towers with 2 cubes in each.
Mary has built 7 towers with 4 cubes in each.
Who has more towers, Rachael or Mary? Your Answer: _____________
How many more towers does she have?

Unit Differentiation and Selection Tom’s father bought 6 pizzas.
Each pizza had 4 slices.
Tom’s mother bought a few more pizzas.
Then, there were 9 pizzas.
How many more slices did Toms’ mother bring?

Mixed Unit Coordination Maria made birthday bags. She wants each bag to have 6 candies. After making 3 bags, she still had 12 candies left. How many bags will she have altogether after putting these 12 candies in bags?

Quotitive Division
There are 28 students in Ms. Franklin’s class.
During reading, she puts all students in groups of 4.
She asked a student (Steve): “How many groups will I make?”
Steve said: “32. Because 28+4 is 32.”
Do you think that Steve is correct? Your Answer: _____________
Why? _________________________________________________

Partitive Division
After an art class, there were 78 crayons out on the tables.
There are 6 boxes for the crayons.
Ms. Brown puts the same number of crayons in each box.
How many crayons would she put in each box?
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TABLE 2
Sample Problems in COMPS program (from Xin et al., 2008)
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Problem type Sample Problem Situations
Equal Groups
Unit Rate unknown A school arranged a visit to the museum in Lafayette Town. It spent a total of $667 buying 23 tickets. How much does each ticket cost?

Number of units (sets) unknown There are a total of 575 students in Centennial Elementary School. If one classroom can hold 25 students, how many classrooms does the school need?

Product unknown Emily has a stamp collection book with a total of 27 pages, and each page can hold 13 stamps. If Emily filled up this collection book, how many stamps would she have?

Multiplicative Compare
Compared set unknown Isaac has 11 marbles. Cameron has 22 times as many marbles as Isaac. How many marbles does Cameron have?

Referent set unknown Gina has sent out 462 packages in the last week for the post office. Gina has sent out 21 times as many packages as her friend Dane. How many packages has Dane sent out?

Multiplier unknown It rained 147 inches in New York one year. In Washington D.C., it only rained 21 inches during the same year. The amount of rain in New York is how many times the amount of rain in Washington D.C.?
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For publications, watch the two slides titled “project related publications” towards the end of the video.
For challenges and sample publications of field tests of the program, please refer to:
Xin, Y. P., Liu, J., Jones, S., Tzur, R., SI, L. (in press). A Preliminary Discourse Analysis of Constructivist-Oriented Math Instruction for A Student with Learning Disabilities. The Journal of Educational Research.